# Axiom:Axiom of Foundation

## Axiom

For all non-empty sets, there is an element of the set that shares no element with the set.

That is:

- $\forall S: \left({ \left({\exists x: x \in S}\right) \implies \exists y \in S: \forall z \in S: \neg \left({z \in y}\right) }\right)$

The antecedent states that $S$ is not empty.

It can also be stated as:

- For every non-empty set $S$, there exists an element $x \in S$ such that $x$ and $S$ are disjoint.

- A set contains no infinitely descending (membership) sequence.

- A set contains a (membership) minimal element.

- The membership relation is a foundational relation on any set.

## Also known as

The **axiom of foundation** is also known as the **axiom of regularity**.

## Sources

- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html